ON SOME IMPROPERLY POSED PROBLEMS FOR QUASILINEAR EQUATIONS OF MIXED TYPEC)

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1 ON SOME IMPROPERLY POSED PROBLEMS FOR QUASILINEAR EQUATIONS OF MIXED TYPEC) BV L. E. PAYNE AND D. SATHER 1. Introduction. In a previous paper [6] the authors considered certain improperly posed initial-boundary value problems for the Chaplygin equation 82u d2u (i.i) ^+h(y) 2 = o. ey+ky)st2 The Chaplygin equation, which arises in the study of transonic flow, is one of the simplest equations of mixed type. It was shown that with appropriate hypotheses on the function «a problem of this type could be stabilized (i.e., the solution made to depend continuously on the data) by restricting the solution to lie in the class of functions whose L2 integrals are bounded by some prescribed constant M. Similar restrictions have previously been shown (see e.g., [2], [7]) to guarantee continuous dependence in certain improperly posed problems for elliptic and parabolic equations. However, little study has up to now been devoted to improperly posed problems for equations of mixed type. For that matter one is rarely able to determine just what is a well set problem for such an equation. In this paper we demonstrate that the convexity arguments which led to stability inequalities for the Chaplygin equation [6] actually can be carried over with some modification to yield stability inequalities and error bounds in improperly posed problems for a wide class of quasilinear partial differential equations of mixed type. We shall be concerned here only with the question of obtaining such inequalities. We leave aside the difficult question of existence. The particular equation which we consider is by no means the most general one amenable to the convexity method. We restrict our attention to a relatively simple equation but one which exhibits most of the troublesome features which would appear in the more general operators. A survey of the literature on improperly posed problems will appear in a forthcoming paper by Payne [5]. In this paper we consider the following initial-boundary value problem for a cylindrical domain 2 = Z)x(0, Y), D is an arbitrary «-dimensional domain with smooth boundary D: Received by the editors July 1, O This research was supported in part by the National Science Foundation Grant No. GP 5882 with Cornell University. 135

2 136 L. E. PAYNE AND D. SATHER [July Problem A. (1.2) Lu = uyyy-(aitu_i)j-f(x, y, u, u%v) = 0 in Q, (L3) u prescribed on Ù D x [0, Y], du u, -r- prescribed on y = 0. Here (ay) is a symmetric matrix function of the variables x = (xx, x2,..., xn) and y. The matrix (ay) is not assumed to be definite or even semidefinite. The repeated index is used to designate summation from 1 to n and uii = du 8xi. The function/ is to be Lipschitz continuous in its last two arguments, i.e., (1.4) \f(x,y,ux,ux y)-f(x,y,u2,u2 y)\ ú Mx\ux-u2\+M2\ux y-u2 y\. We assume that there exists a nonnegative constant S such that the matrix (Fi;) with elements (1.5) Fy = ^ay-2(m2 + 8)ay is negative semidefinite in Q. Here A/2 is given by (1.4). We shall say that u belongs to Ji if the condition (1.6) f u2dxdy g AT2 Ja is satisfied for some prescribed constant M. In the following section we will develop a priori inequalities which for solutions u of Problem A that belong to J( may be used to obtain uniqueness, continuous dependence on the data and L2 bounds on compact subdomains of O. 2. The a priori inequalities. We derive the basic inequalities in a form which yields immediately the desired L2 bounds; by specialization, these inequalities imply also the properties of uniqueness and continuous dependence on the data. Let us approximate m by a function <p which is piecewise C2 in O, Cx in O and such that u = <p on Q. We set Qy = D x (0, y), 0 < y ^ Y, and (2.1) v = u 9. Let Í20 denote the intersection of Q. and the initial plane y = 0 and let ^ denote the functional (2.2) &(v, y) = log F(v, y) + co2, 0 < y < Y, (2.3) a = (y0+y)-a, «> 0, and F(v,y)= v2dxdr + (Y-y)\ v2dx+kx[ v2 dx (2.4) +k2 f O^i + ir'jax-l-ia f [Lpfdxdy JCl0 JC1 = f v2dxdr,+ Q2.

3 1967] SOME IMPROPERLY POSED PROBLEMS 137 Here the constants a, c, and kt in (2.2) through (2.4) are to be determined. The positive constant y0 is introduced so that o ranges over a finite interval. We note that Q in (2.4) involves only data terms of u <p and, hence, for fixed k it will be small if the data of <p approximate the data of u sufficiently well. Our a priori inequalities are a consequence of the following convexity argument. If we can show that the functional F is a convex function of a then an application of Jensen's inequality yields (2.5) er if v2dxdr] + Q2\ IeeE2 v2dxdy+q2\j {ec o2qf«-»««.-2>, (2.6) zz = (Y+y0ya and a0 = yô«. Since m is to satisfy (1.6) and <p is at our disposal we may assume without loss of generality that (2.7) f v2 dx dy = Ml In addition, since Q involves only data terms we have (2.8) Q2 = vm2o v is a computable constant. Therefore (2.9) f v2dxdr +Q2 S ec<ffo-">«'-»{(l+v)af }(<'o-''>'«'o-»ßa«7-»/(a0-s)> This last inequality yields the desired L2 bounds for u on compact subdomains of O. Let us now show that J5" is a convex function of a. We note that (2.10) F2JF" = FF" - (F')2 + 2cF2, the prime denotes differentiation with respect to a. Therefore, in order to establish the convexity of J5", it is sufficient to show that the right-hand side of (2.10) is positive. Since (2.11) F = -«-»»-"^"ï dy (2.12) F" = a-2(l+a)a-<1 + 2a)/a^+a-2a-2<1 + a>'*^ we have (2.13) -OT<1 + *F = 2 í v(x,r )v_y(x,ri)dxdr (2.14) a2a2(1 + a)/af" = 2(1 +a)alla Í w,y dx dv + 2 v2ydx d-q + 2 vvivdx +2 vv<yydxdr. JCly Jn0 JCiy

4 138 L. E. PAYNE AND D. SATHER [July The last term in (2.14) may be rewritten as (2.15) and, hence, jay vvivvdxdr)=\ v{(a,jv,t)j-l<p+f(u)-f(<p)}dxdri Jav = {-aav,ivj-vl<p+v[f(u)-f(<p)]}dxdri Jav a2ct2(i+«)/«f» = 2(1 + < )olla f wty dxdq+4 ( v2y dx dr, Jdy Jß (2.16) + 2 f {v[f(u)-f(<p)]- vdp} dxdr +2 ja Jn0 vv,y dx 2 (v^y+aijvjvj) dxdr. Jay An upper bound for the last term in (2.16) is obtained as follows. We note that for 0</<j» 0 = 2 (t ri)v,y{v,vy-(aitv^tt+l<p-[f(u)-f(9)]}dxdy = 2 f (t- r,)v,y{l<p - [/(«)-/(?)]} dx dr, + (v2y+oijvjvj) dxdrj t (v2y+atjvwivj) dx JClt Jci0 - (t-v)au.t,v.ivjdxdr. Jat Let us now assume that the constant 8 in (1.5) is positive. An application of the arithmetic-geometric mean inequality, and (1.4), (1.5), and (2.17) imply (2.18) Ä) g J0»)+2(Ma+8»(t) (2.19) J(t)=i (.t-tufâ+atfljvjdxdn, Ja, (2.20) I(y) = y \ (v2y+aitvivt)dx+2mxy\ \vv J dx dr + -r f [Up]2 dx dr,. Jn0 Jn ' 4o Ja Integrating (2.18) from 0 to y we obtain (2.21) f (v2y + aip fi t) dx dr, S 7(j)e2(M2+«". Jav

5 1967] SOME IMPROPERLY POSED PROBLEMS 139 Substituting (2.21) into (2.16) and using (1.4) we have for some computable constants bi that a2a2(1 + a)/a[ff"-(f')2] ^ 2(1 + cofa1'«f vv,y dx dt]+as2 + 4g2 f v2y dx dr (2.22) F-lbx v2 dx+b2 v2ydx + b3 vtfvtidx I Jo0 Joo J^o + bt [L<p]2 dx dy + b5 v2 dx dt] + 4b6 \vvtv\ dx drn Jo Jo Jo,, (2.23) S2 = ( f v2 dx dr ]( v2y dx dr ]-( vv,y dx dv) \Jny /\Jnv I \Jnv I Since (2.24) vvyv dx dn ^ vviv dx drn v2 dx JCly Jo JOq and (2.25) \vv,y\ dxdrj = S+\ vvtydxdn, Jciy Joy it follows from (2.22) that a2a2<1+«>'«[ff"-(f')2] ^ 2F[(l + a)c71/a-2è6] f vv y dxdr\ + (2S-b6F)2-b26F2 (2.26) r / c -FÏ [bx + 2(l +a)o1'a] v2dx + b2\ v% dx I Jo0 Ja0 + b3 VjVjdx + bi [Dp]2 dx dy + bs v2 dx do Jo0 Ja Jciy If we now choose 1 + a=2be(y+y0) then the coefficient of the first term in (2.26) is nonnegative and, hence, a2a2(i + a>/af2jf» ^ F2[2ca2(F+j0)-2(1 + a)-» ] (2.27) -f([í>i+2(1 + a)/y0] f v2dx + b2[ v2y dx + b3 vavtidx+bi [Dp]2 dxdy+bs v2 dx dr Jo0 ' JO JQy Clearly the constants klt k2, k3 in Q, and c can be chosen so that the right-hand side of (2.27) is positive. In fact for any positive kx, k2, and k3 we need only take

6 140 L. E. PAYNE AND D. SATHER [July c large enough so that the first term in (2.27) is the dominant one. Thus it is possible to choose the constants a, ku and c so that IF is a convex function of a. We have established the following result. Theorem 1. Suppose that S>0 i«(1.5) and the matrix (Bi,) is negative semidefinite. Let u satisfy (1.2), (1.3), and (1.6), and let v, a, Q, 2, o0, and M0 be defined as in (2.1), (2.3), (2.4), (2.6), and (2.7), respectively. Then there exists a computable constant K such that (2.28) f v2dxdn ^ TÍM^o-^'o-sg^-s/í'o-s), 0 < y < Y. JCiy An immediate consequence of the above discussion is Theorem 2. Suppose that the constant 8 «(1.5) is nonnegative. If the matrix (Z?y) is negative semidefinite then any solution of Problem A that is in the class Jt of functions whose L2 integrals over Q are bounded by some constant M will depend continuously in L2 on the Cauchy data. Let us remark that if, in particular, the matrix (ai;) is independent of y, 8=0 and the function fis independent of u y then (Z?i;) = 0. Moreover, the requirement on the matrix (Btj) is a "best possible" condition in the sense that one can give examples Theorem 2 does not hold when this condition fails to be satisfied [6]. Finally, the restriction S>0 in Theorem 1 can be replaced by 8^0 if one imposes additional smoothness requirements on <p (see [5], [6]). As an interesting specific application of these results we let D be the rectangular parallelepiped {0<x<X,0<y< Y,0<t<T} and consider the following Cauchy problem for the wave equation : Problem A. (2.29) mfj/1, = u tt u>xx in Q, (2.30) u prescribed on x = 0, x = X, t = 0, and t = T, (2.31) u, 7T- prescribed on y = 0. Then any solution u of Problem A' that is in the class Jt will depend continuously in L2 on the Cauchy data (compare also [1], [3], and [4]). Bibliography 1. J. R. Cannon, A Cauchy problem for the wave equation for a space-like region, Le Matematiche 16 (1961), F. John, A note on "improper" problems in partial differential equations, Comm. Pure Appl. Math. 8 (1955), , Continuous dependence on the data for solutions of partial differential equations with a prescribed bound, Comm. Pure Appl. Math. 13 (1960), L. Nirenberg, Uniqueness in Cauchy problems for differential equations with constant leading coefficients, Comm. Pure Appl. Math. 10 (1957),

7 1967] SOME IMPROPERLY POSED PROBLEMS L. Payne, On some non-wellposed problems for partial differential equations, Proc. Sympos. Numerical Solutions of Nonlinear Differential Equations, pp , Math. Res. Center, U.S. Army, Univ. of Wisconsin, Madison, Wiley, New York, L. Payne and D. Sather, On some improperly posed problems for the Chaplygin equation, J. Math. Anal. Appl. (to appear). 7. C. Pucci, Sui problemi di Cauchy non "ben posti," Rend. Accad. Naz. Lincei 18 (1955), Cornell University, Ithaca, New York